A081386 - OEIS

%I #16 Dec 18 2016 01:35:00

%S 1,2,1,3,1,3,3,4,4,4,5,5,4,3,5,7,6,7,7,8,9,9,6,7,7,7,8,11,12,11,11,11,

%T 12,12,12,13,13,13,11,13,12,14,13,13,15,14,14,14,15,16,16,16,17,19,18,

%U 17,18,19,18,19,18,18,18,20,18,21,22,20,20,20,20,20,20,19,21,21,24,23

%N Number of unitary prime divisors of central binomial coefficient, C(2n,n) = A000984(n), i.e., number of those prime factors in C(2n,n), whose exponent equals one.

%H T. D. Noe, <a href="/A081386/b081386.txt">Table of n, a(n) for n=1..2000</a>

%F a(n) = A056169(A000984(n)).

%e n=10: C(20,10) = 184756 = 2*2*11*13*17*19; unitary-p-divisors = {11,13,17,19}, so a(10)=4.

%t Table[Function[m, Count[Divisors@ m, k_ /; And[PrimeQ@ k, GCD[k, m/k] == 1]]]@ Binomial[2 n, n], {n, 50}] (* _Michael De Vlieger_, Dec 17 2016 *)

%o (PARI) a(n) = my(f=factor(binomial(2*n, n))); sum(k=1, #f~, f[k,2] == 1); \\ _Michel Marcus_, Dec 18 2016

%Y Cf. A000984, A067434, A081387-A081389, A034444, A048105, A056170, A056169.

%K nonn

%O 1,2

%A _Labos Elemer_, Mar 27 2003